TY - JOUR

T1 - Unitary part of a contraction

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information:
This research was supported by the National Science Council of the Republic of China under NSC 96-2115-M-008-006 (H.-L. Gau) and NSC 96-2115-M-009-013-MY3 (P.Y. Wu). The second author was also supported by the MOE-ATU project and has had the contents reported in the 3rd International Workshop on Matrix Analysis and Applications at Lin An, Zhejiang, China on July 11, 2009.

PY - 2010/6/15

Y1 - 2010/6/15

N2 - For a contraction A on a Hilbert space H, we define the index j (A) (resp., k (A)) as the smallest nonnegative integer j (resp., k) such that ker (I - Aj * Aj) (resp., ker (I - Ak * Ak) ∩ ker (I - Ak Ak *)) equals the subspace of H on which the unitary part of A acts. We show that if n = dim H < ∞, then j (A) ≤ n (resp., k (A) ≤ ⌈ n / 2 ⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with {norm of matrix} An - 2 {norm of matrix} = 1 and {norm of matrix} An - 1 {norm of matrix} < 1, and (3) n is even and A = U ⊕ A′, where U is unitary on a one-dimensional space and A′ is of class Sn - 1).

AB - For a contraction A on a Hilbert space H, we define the index j (A) (resp., k (A)) as the smallest nonnegative integer j (resp., k) such that ker (I - Aj * Aj) (resp., ker (I - Ak * Ak) ∩ ker (I - Ak Ak *)) equals the subspace of H on which the unitary part of A acts. We show that if n = dim H < ∞, then j (A) ≤ n (resp., k (A) ≤ ⌈ n / 2 ⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with {norm of matrix} An - 2 {norm of matrix} = 1 and {norm of matrix} An - 1 {norm of matrix} < 1, and (3) n is even and A = U ⊕ A′, where U is unitary on a one-dimensional space and A′ is of class Sn - 1).

KW - Completely nonunitary part

KW - Contraction

KW - Norm-one index

KW - S-operator

KW - Unitary part

UR - http://www.scopus.com/inward/record.url?scp=76749103447&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2010.01.040

DO - 10.1016/j.jmaa.2010.01.040

M3 - 期刊論文

AN - SCOPUS:76749103447

VL - 366

SP - 700

EP - 705

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -